68 research outputs found
On the quantum chromatic number of a graph
We investigate the notion of quantum chromatic number of a graph, which is
the minimal number of colours necessary in a protocol in which two separated
provers can convince an interrogator with certainty that they have a colouring
of the graph.
After discussing this notion from first principles, we go on to establish
relations with the clique number and orthogonal representations of the graph.
We also prove several general facts about this graph parameter and find large
separations between the clique number and the quantum chromatic number by
looking at random graphs.
Finally, we show that there can be no separation between classical and
quantum chromatic number if the latter is 2, nor if it is 3 in a restricted
quantum model; on the other hand, we exhibit a graph on 18 vertices and 44
edges with chromatic number 5 and quantum chromatic number 4.Comment: 7 pages, 1 eps figure; revtex4. v2 has some new references; v3 furthe
small improvement
Quantum state-independent contextuality requires 13 rays
We show that, regardless of the dimension of the Hilbert space, there exists
no set of rays revealing state-independent contextuality with less than 13
rays. This implies that the set proposed by Yu and Oh in dimension three [Phys.
Rev. Lett. 108, 030402 (2012)] is actually the minimal set in quantum theory.
This contrasts with the case of Kochen-Specker sets, where the smallest set
occurs in dimension four.Comment: 8 pages, 2 tables, 1 figure, v2: minor change
Graph-theoretical Bounds on the Entangled Value of Non-local Games
We introduce a novel technique to give bounds to the entangled value of
non-local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of
entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost
of a factor of two in the total communication, Alice could just as well have
given the two possible replies without listening and have Bob select which
applies to him. Motivated by a conjecture stating that this form of "round
elimination" is impossible in exact quantum communication complexity, we study
the orthogonal rank and a symmetric variant thereof for a certain family of
Cayley graphs. The orthogonal rank of a graph is the smallest number for
which one can label each vertex with a nonzero -dimensional complex vector
such that adjacent vertices receive orthogonal vectors.
We show an exp lower bound on the orthogonal rank of the graph on
in which two strings are adjacent if they have Hamming distance at
least . In combination with previous work, this implies an affirmative
answer to the above conjecture.Comment: 13 page
Characterization of Binary Constraint System Games
We consider a class of nonlocal games that are related to binary constraint
systems (BCSs) in a manner similar to the games implicit in the work of Mermin
[N.D. Mermin, "Simple unified form for the major no-hidden-variables theorems,"
Phys. Rev. Lett., 65(27):3373-3376, 1990], but generalized to n binary
variables and m constraints. We show that, whenever there is a perfect
entangled protocol for such a game, there exists a set of binary observables
with commutations and products similar to those exhibited by Mermin. We also
show how to derive upper bounds strictly below 1 for the the maximum entangled
success probability of some BCS games. These results are partial progress
towards a larger project to determine the computational complexity of deciding
whether a given instance of a BCS game admits a perfect entangled strategy or
not.Comment: Revised version corrects an error in the previous version of the
proof of Theorem 1 that arises in the case of POVM measurement
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
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